具有齐次、方向反局部主象征的非局部薛定谔方程的Calderón问题
On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols
摘要：
在这篇文章中，我们考虑$α$稳定的，椭圆型非局部算子的核心可能只支持在锥上，并满足\emph{directional antilocality}的结构条件，在\cite{I86}。我们认为这些运营商的“域依赖的运营商”，并在几个，适应函数空间的狄利克雷问题。这种提法可以避免自然的“仪表”，否则将不得不考虑在相关的反问题的研究。利用这些运营商的方向反局部性，我们补充调查的\emph{direct problem}与无限的数据和单测量的唯一性结果相关联的\emph{inverse problems}。在这里，由于唯一的方向反局域性，新的几何条件出现在测量域。我们讨论了对称和一类特殊的非对称非局部椭圆算子的情形，并对比了相应的正问题和反问题的结果。特别是只有“单边算子”的新现象出现在正问题和反问题中：例如，它可以在涉及局部和非局部数据的数据空间中研究问题，唯一的连续性可能一般不成立，并对反问题的测量集产生进一步的限制。
英文摘要：
In this article we consider direct and inverse problems for $α$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of \emph{directional antilocality} as introduced in \cite{I86}. We consider the Dirichlet problem for these operators on the "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the \emph{direct problem} with infinite data and single measurement uniqueness results for the associated \emph{inverse problems}. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only "one-sided operators" new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.
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